Positive-definite block matrix with constant block sums

Question

Given two natural numbers $n$ and $m$, suppose that $A$ is an $nm\times nm$ real nonnegative matrix. Seeing $A$ as a block matrix where each block has size $m\times m$, suppose that the sum of the entries in each block is $1$. Can $A$ be positive-definite? Notice that making $A$ positive-semidefinite is easy, for example we can let $A=\frac{1}{m}J_n\otimes I_m$. Also, notice that for $m=1$ the answer is no, as the only matrix satisfying the constraints is the all-one matrix.

Answer 1

It cannot possibly be positive definite because $x^TAx=0$ when $$ x=(\underbrace{1,\ldots,1}_{m},\underbrace{-1,\ldots,-1}_{m},0,\ldots,0)^T. $$